Instanton 2-torsion and Dehn surgeries

2025/08/05 by Li, Zhenkun, Ye, Fan · 2 citations
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.2508.03394

Abstract

In our earlier work on 2-torsion in instanton Floer homology, we considered only integral surgeries on a knot K⊂ S³ and showed that the absence of 2-torsion forces K to be fibered. The present paper extends the result to all rational surgeries. We prove that if the framed instanton homology I^\sharp(S³_r(K);ℤ) is 2-torsion-free for some r∈ ℚ₊, then K is an instanton L-space knot and r>2g(K)-1. Leveraging this 2-torsion perspective, we also obtain new small-surgery obstructions: If either S³₅(K) or S³_11/2(K) is SU(2)-abelian, then K must be the unknot or the right-handed trefoil. This result sharpens the small-SU(2)-abelian surgery theorems of Kronheimer--Mrowka, Baldwin--Sivek, and Baldwin--Li--Sivek--Ye.

Citations

• 2-torsion in instanton Floer homology
• Surgery Exact Triangles in Instanton Theory
• Knot surgery formulae for instanton Floer homology II: applications
• Framed instanton homology and concordance, II
• Knot surgery formulae for instanton Floer homology I: the main theorem
• SU(2) representations and a large surgery formula
• Decomposing sutured monopole and Instanton Floer homologies
• On the tau invariants in instanton and monopole Floer theories
• Ribbon homology cobordisms
• Knot homologies in monopole and instanton theories via sutures
• Knots, sutures and excision
• The contact invariant in sutured Floer homology
• Dehn surgery, the fundamental group and SU(2)
• Monopoles and lens space surgeries

Cited by

• Singular instanton homology of dual knots
• Instanton dimensions of knot surgeries over arbitrary fields

Related

• 2-torsion in instanton Floer homology
• Framed instanton homology and Frøyshov's invariant
• Knot cobordism, torsion order and framed instanton homology
• Knot surgery formulae for instanton Floer homology II: applications
• Framed instanton homology and concordance, II